|
How are we to classify the straight line? Shall we deny that
it is a magnitude?
The suggestion may be made that it is a qualified magnitude. May
we not, then, consider straightness as a differentia of "line"?
We at any rate draw on Quality for differentiae of Substance.
The straight line is, thus, a quantity plus a differentia; but it
is not on that account a composite made up of straightness and
line: if it be a composite, the composite possesses a
differentiae of its own.
But [if the line is a quantity] why is not the product of three
lines included in Quantity? The answer is that a triangle
consists not merely of three lines but of three lines in a
particular disposition, a quadrilateral of four lines in a
particular disposition: even the straight line involves
disposition as well as quantity.
Holding that the straight line is not mere quantity, we should
naturally proceed to assert that the line as limited is not mere
quantity, but for the fact that the limit of a line is a point,
which is in the same category, Quantity. Similarly, the limited
surface will be a quantity, since lines, which have a far better
right than itself to this category, constitute its limits. With
the introduction of the limited surface- rectangle, hexagon,
polygon- into the category of Quantity, this category will be
brought to include every figure whatsoever.
If however by classing the triangle and the rectangle as qualia
we propose to bring figures under Quality, we are not thereby
precluded from assigning the same object to more categories than
one: in so far as it is a magnitude- a magnitude of such and such
a size- it will belong to Quantity; in so far as it presents a
particular shape, to Quality.
It may be urged that the triangle is essentially a particular
shape. Then what prevents our ranking the sphere also as a
quality?
To proceed on these lines would lead us to the conclusion that
geometry is concerned not with magnitudes but with Quality. But
this conclusion is untenable; geometry is the study of
magnitudes. The differences of magnitudes do not eliminate the
existence of magnitudes as such, any more than the differences of
substances annihilate the substances themselves.
Moreover, every surface is limited; it is impossible for any
surface to be infinite in extent.
Again, when I find Quality bound up with Substance, I regard it
as substantial quality: I am not less, but far more, disposed to
see in figures or shapes [qualitative] varieties of Quantity.
Besides, if we are not to regard them as varieties of magnitude,
to what genus are we to assign them?
Suppose, then, that we allow differences of magnitude; we commit
ourselves to a specific classification of the magnitudes so
differentiated.
|
|